I struggled with this for a long time as an elementary and middle school math teacher. Eventually, I realized two things:
1) Character traits are often more important to success than particular knowledge/skills. Grit is something that you learn by working through hard problems.
2) Interesting != Useful. The best teachers help students see what’s intrinsically interesting about a subject. Once students are engaged, the question, “When are we going to use this?” doesn’t even arise.
A teacher who has thought deeply about this in math education is Dan Meyers: http://blog.mrmeyer.com/ Although he focuses on high school education, I think his ideas would work at the college level as well.
Dghhdfbj khffhjsays:
Character traits are inborn, or at least beyond our control. You’re simply rewarding or punishing students for being who they are. You should kill yourself
Joel Chansays:
Two responses:
1) Yup, we probably overrate how much good learning “useless” subjects will do to students, and probably overestimate the harm that happens by doing it badly (or perhaps at all).
2) Transfer is a slippery cognitive phenomenon. Oftentimes it does indeed look like it almost never happens, except when there’s much overlap in situational features. On the other hand, there’s something to be said about the alternative “preparation for future learning” view of transfer, where transfer isn’t just literally using the knowledge/skill bits you learned from before in a new situation, but your learning in that prior situation having better prepared you to ask the right questions and use the resources in your new situation to achieve understanding/solve your new problem.
3) From the perspective of creative problem solving, I think it is helpful to cultivate interests and skills in a heterogeneous set of domains. It can give you a wider range of representations and solution approaches when you’re faced with a difficult problem. However, I agree with you that this needs to be intrinsic-interest-driven.
Even if transfer learning would not work at all, we should teach people a wide array of things in school. Many young people don’t want to learn things that they won’t use in their future job. But they don’t know what job they will be able to get in the future (the job market is complicated and unpredictable), or what skills are useful there. Further, they won’t know if they want to work with skill/subject X until they have learned enough about it to understand.
Finally, even if transfer learning does not work at all, knowing many different things will certainly aid a particular type of creativity, where you come up with new facts or new things by combining disparate previously known facts.
Marcosays:
In Italy, we have a high school (Liceo Classico) where we study Latin and Ancient Greek. I attended it and I finished my course with top marks (so I know what I am talking about). People say Latin and Greek are useful to form your mind and, as evidence of this, they say that former Liceo Classico students perform very well at university. But… they do not say that these kids are always good students also before attending Liceo Classico (which is actually a difficult school) and often their parents have a higher education and sometimes their families are wealthy. Thanks for your post.
I’d go along if I thought that we picked teaching topics based on what people are likely to need later. I don’t think it works this way. For example, we teach advanced trigonometry, but rarely teach probabilities in high school.
Few will ever need anything from trigonometry but the basics. Heck! I am a college professor with a Ph.D. in engineering mathematics and I don’t recall the last time I had to use a trigonometric identity, let alone some of the crazy ones. And if I needed to do trigonometry today, I would probably not bother with the crazy identities. Is it our belief that people (what proportion… 0.1%?) who will need to know these advanced trigonometric identities won’t be able to learn it then? Are we so arrogant as to think that all should learn it so that the few who ever need will know them?
I don’t think it is the logic we use. We impose trigonometry to all students on the basis that even if they never use it, it will be good for them to do some crazy mathematics.
(And also because they will need to differentiate cos^2(x) later if they take calculus… but why exactly do they need to differentiate cos^2(x)? Hardly a useful skill for most people…)
(BTW the net effect of these teaching strategies might well be negative as it can be shown that most students finish high school with a hatred of mathematics. So if they do need trigonometry later, they may reasonably try to avoid it for fear of boredom. Hardly what we set out to do…)
Andrew McMillansays:
I’d like to read more of the research on the subject of transfer. For example do the authors discover that it never occurs? Or that ‘on average it has little effect’? If the first, then I believe it is still valuable because (for example) you cannot say in advance that (e.g.) trigonometry will have no use to someone studying encryption techniques. Or that studying psychology will have no value to a programmer.
While in the general case there may well be no cross-pollination across such fields, in some specific cases the crossover may produce startlingly original insight, and I think this is where huge value can lie.
In my own experience of 30-odd years of programming I have found very valuable crossover from my previous career as an electronics technician. And in that earlier career I recall being significantly surprised as I drew on my high-school math experience with imaginary numbers – surely useless knowledge indeed!
Chandsays:
I think while you are making the point (and quite well!) for one way the other side is quite important. It is confusing that learning the “other” things would help in critical thinking, it is not necessary. It is not really clear what helps in critical thinking, indeed had that been know then that is what would be taught. My understanding is that when one does develop or starts to develop critical thinking – then all the “useless” things whether done for fun (more like to help) or imposed help and help a lot in a non-tangible non-obvious way.
Paulsays:
I’d argue there are useful transfers from trigonometry. It’s a helpful tool for approaching Non-Euclidean geometries, which may end up being the universe’s geometry and at the very least define distances on Earth.
Additionally, the trigonometric functions encapsulate discontinuous and periodic functions rather nicely. If you can understand function composition and calculus involving trig functions, you’ve gone a long way to understanding function composition/calculus in general. And I’ve certainly found ample opportunities to reason about how different processes would combine or change over time.
Vector space models can be great for reasoning or calculating on complex objects as a class, rather than reasoning from the properties of each variable explicitly.
You certainly don’t need to be able to recite the trig identities, but I’ve found the intuition about angles has stuck with me and come in handy. On the same tokens I’m not an economist but a basic handle on the workings of an economy help me understand the connections of what I learn about the world around me.
Suresh Venkatsays:
I think that arguing about why (say) trigonometry might be useful misses the point. We can construct a plausible rationale for all knowledge being useful – after all, that’s why it first started being developed !
The point is that given the limited amount of time that students in school have, what are the MOST important things they should learn from any topic. For example, there’s a really strong argument for probability based on just the sheer number of places where a misunderstanding of probability can cause problems. This is a feature of our modern data-driven world, and wouldn’t have been true 100 years ago. And while geometry is one of my favorite topics, I don’t necessarily see the need to use it as “the way to expose students to logical reasoning”. In fact, a study of discrete mathematics or graph theory might do the same, and is easily relatable to student experiences.
Paulosays:
I’d be willing to accept the meat in your argument — not everything you learn is useful per se, and given our limited time, we’d do well to try to be selective, if we can. But I’d say that even useless learning has its purpose in keeping the brain “alive”. It’s a bit like gymnastics/exercising: not every kind is useful for you, and some of them may even cause harm if done wrong. But in general any exercise is better than doing nothing. As for the brain, though not totally proven it appears than making it “exercise” would be good to prevent cognitive degeneration in old ages (though that is mostly unrelated to learning at early ages).
And I’d take issue with you example: mathematics *is* useful/important for everybody. Though you later qualify it in one comment: from among branches of mathematics I agree to stress probability, statistics and basic calculus, and less other areas (such as differential calculus or trigonometry). Numerical literacy for everybody would help to avoid people getting tangled in mortgages they don’t understand, or fooled by bogus statistics (the famous “How to lie with statistics” book should probably be compulsory 🙂
Muigaisays:
There are learning experiences that can leave a lasting impression without imparting anything useful. Consider for example an engineering student who takes an art history class. Let’s not make a fetish of utilitarianism.
Daniel Haransays:
Can anyone here argue in good faith that we should be privileging trig over probabilities in high school?
As to what is useful, the goal is to form critical thinkers that can be effective workers and good citizens. I’m pretty sure we’re failing at all of those tasks right now.
‘I consider that a man’s brain originally is like a little empty attic, and you have to stock it with such furniture as you choose. A fool takes in all the lumber of every sort that he comes across, so that the knowledge which might be useful to him gets crowded out, or at best is jumbled up with a lot of other things, so that he has a difficulty in laying his hands on it. Now the skillful workman is very careful indeed as to what he takes into his brain-attic. He will have nothing but the tools which may help him in doing his work, but of these he has a large assortment, and all in the most perfect order. It is a mistake to think that little room has elastic walls and can distend to any extent. Depend uon it – there comes a time when for every addition of knowledge you forget something that you knew before. It is of the highest importance, therefore, not to have useless facts elbowing out the useful ones.’
Stefanosays:
I attended Liceo classico in Italy like Marco. I’ve to confess that for me studying Latin and ancient Greek was fun, and what I learned is still useful for me today.
When learning languages (dead or alive, it doesn’t matter) there is some sort of critical mass: when you have acquired a good knowledge of say four of five different languages, everything suddenly becomes easier. The same holds for math, up to a certain degree. The key point is that you need a deep understanding of the subject; today’s skill oriented learning is not enough to trigger the advantages of being a polymath.
Dominic Amannsays:
Stating that learning useless stuff is useless is pretty much beggin the question. Of course it is – by definition.
Learning Latin (as I did) is valuable in learning formal grammar rules (which are seldom taught when one learns english), so learning other languages becomes easier, and one has a language to use for understanding other grammars (including computer languages). It is also extremely valuable in learning French, Spanish and Italian – all of whose newspapers I can read.
Another example is Music. One can argue that music is irrelevant for a scientist, however, there are numerous studies linking music to better performance in math.
Learning the “classics” gives me an appreciation of how little is new in terms of humanities – a broad perspective of the deep problems of mankind, and lessens my hubris as a scientist.
In the workplace, a sales education has helped me be a better programmer by enabling a much better communication channel between our programming department and the front end of our company.
Did I know that any of these things would be useful 30 years ago when I studied them? Not at all. Does anyone really know what they will be doing in 10, 20 or 30 years – and what studies might either directly or peripherally be useful? I sincerely doubt it.
Marcosays:
@Stefano, it was fun to me, too. 🙂
I was just saying that the importance of learning difficult things is overestimated.
You say that after a degree everything becomes easier.
I do not completely disagree with you, but are you sure that it is not because you become more familiar to a subject and not smarter?
I do not say that this is a negative thing. Anyway, you improve your understanding. For this reason, I am a believer in deep and vary knowledge. Perhaps, this is what Latin and Greek gave me.
This seems to confuse learning and teaching (or learning and education).
I’m sure that sitting in a course and having a subject matter (say, projective geometry) poured into your head doesn’t help you with unrelated subjects in the future.
But self-directed learning, where the student is actually active rather than a passive receptacle, I think does help, Figuring out projective geometry for yourself not only teaches you the subject, it is practice in how to figure out complex things.
The real subject matter of an education should be learning how to learn.
Now that’s an entirely different matter — and I for one think it’s the most important factor in learning.
I went to college for comp sci in my late twenties to realize a life long a dream of being a professional java developer …
Yet I had kids — no money — and a lot of bills so I had to put it on the back burner —
A few years later I was working for a company that owned 33,000 web properties. I was in a sales position and wanted to leave because I wasn’t making nearly what I expected to make.
When I gave my notice the CEO looked at my background {which is checkered} and my college scores {pretty good} and asked if I wanted to work as an SEO … I didn’t know what that position really was but I had a pretty good idea …
After a few months at the new position I was speaking with one of the lead developers and told him
that my life’s goal was actually to be a developer … So, I asked him how I could move into application development … He just looked at me and didn’t say anything …
I kinda’ thought I was after his job … so I dropped the subject but a couple days later
he dropped a 1200 page book on ColdFusion in front of me and said, “We need developers – if you want to be a developer – be a developer … ”
That was 2006 … I’ve read a lot of 1200 page books since then and still doing what I love …
But other than a fantastic couple of years I basically been doing this because I love it …
Cause’ I’m definitely not getting rich … But learn? I study non-stop …
I have to imagine that it would be very difficult to test college students for transfer, considering the problems even measuring what they learn. I will say a couple things about the utility of various math topics:
1. If you were an electrical engineer, you might have had more use for trig.
2. Trig is, in may respects, easier to understand and teach than probability. Considering how low paid K-12 teaching is, one shouldn’t be surprised that it generally attracts people who avoided math like the plague in college. Asking many K-12 teachers to teach probability is likely a recipe for disaster. In any event, it’s in geometry, not trigonometry, where you learn logic and proofs.
3. There’s transfer of knowledge and there’s transfer of ways of thinking and of work skills. Math is certainly as useful as introductory CS for learning such things (maybe more so, considering how poorly Intro to CS is taught many places). Clearly, there are some things valuable to the practice of developing software that one can learn while learning math that one won’t learn, or learn as well, in CS0 or CS1. Like systematic, precise thinking.
Finally, while many of these subjects might not make one a better computer scientist, it might make one a better human being.
There are definitively people who need to know the identity sin(2u) = 2 cos(u) sin(u). But it is a good bet that it is not much more than 1% of all people. Maybe learning this identity makes you a better human being. Maybe.
Paulsays:
I certainly didn’t remember that identity. But there’s so much to sin(2u)=2 cos(u)sin(u)!
Just to start, imagine sin(u). sin(2u) is going to compress that into half the period. That means twice the zero’s. And of course, cos and sin have different zeroes and the same period, so multiplying them would also give you double the zeroes. But isn’t it interesting that the zeroes of cos and the zeroes of sin correspond to alternating zeroes of sin(2u)? And here we can start to see how the fact that the angles in a triangle sum to pi/2 and the definitions of sin and cos in terms of ratios of sides leads to sin(x)=cos(pi/2-x) and that leads to the even spacing of the zeroes. I could go on about where the 2 comes from…
That reasoning uses isomorphisms between a graph, an equation and a definition about angles and triangle sides. What was easy to state in an equation was easy to reason about with a mental graph which added insight to a geometry. I’m convinced that intuition about isomorphisms are hugely valuable for general reasoning, and that being able to switch between them in trig transfers to the isomorphisms in other math fields (e.g. slope, derivative and velocity). Over a decade after a trig course I don’t know many identities but can still reason about what they say. And there are isomorphisms entering and leaving trig: from that graph I could start thinking about integrals to get areas, from the area I could think about distance. And in the other direction the triangle could be the angular distance between two documents I’m thinking about in terms of a hyperspace embedding.
I will agree that the way we usually teach children is to just have them memorize that statement as a fact. But I think there’s very little worth memorizing by rote, and very little not worth finding isomorphisms to other portions of your collected knowledge.
@Daniel: It might be more than 1%, but even if it wasn’t, would you be able to tell which 1% before kids even finish high school, knowing that those who didn’t would be highly unlikely to ever become engineers?
Almost everyone needs to understand the basics of probabilities. Enough to understand a throw of dice. Yet we do not teach anything about probabilities in high school (at least where I live).
You know how I learned probabilities? By playing D&D. I had to figure out whether 2D6 was better than 1D10. I just ran the numbers and came up with an answer.
How many kids can figure out the probability of getting a 7 with 2D6? Probably only the top 5%.
It is pretty pathetic.
I know I sound very utilitarian and even maybe a tad red neck, but I think we should focus on teaching useful stuff.
Dominic Amannsays:
Was it Alfred the Great vs. William the Conqueror – 1066?
Unfortunately you (and I) mostly had bad history teachers.
There could be an entirely different reason for why there appears to be no transfer and indeed not a small amount of anger about “useless stuff”.
This comes back to my personal theory that about 15% of teachers are very good, 70 percent are mediocre and 15% are downright awful. We each have a decent teacher of some subject or other in our formative years. We also most likely have a terrible teacher of some other subject in those formative years. We decide, based on these two instances, that one subject is great and will lead towards our future studies and careers, and the other one is useless.
I argue that it is the emotional bias that prevents any skill transfer, mostly because in a class we actively dislike or feel we suck at, we DON’T LEARN ANYTHING – so there is nothing to transfer. If we love a class and are good at it (usually the same thing), that knowledge transfers into EVERYTHING.
jason braswellsays:
I agree that there is almost zero transfer between skill set training. It’s simply what the research shows. (I even think the French/Spanish connection is probably weak or non-existent.)
Furthermore, given that there’s no transfer and that most of what is learned is forgotten, the vast majority of classroom time is wasted.
For example, pretty much every moment spent in history class is a total waste. There is no “skill” transfer, and nobody remembers any of the information anyway. As proof, perform this simple test. Go up to 10 college-educated people and ask them who were the combatants in the Battle of Hastings. My guess is *maybe* one of them will know. Probably 0.
It is true, but as Steve Jobs noted:
“A lot of people in our industry haven’t had very diverse experiences. They don’t have enough dots to connect, and they end up with very linear solutions, without a broad perspective on the problem.”
It is very hard to tell what knowledge is useful and what is lumber.
Ann Goethalssays:
Hello,
I would just like to point out that every single one of the respondents learned how to write…..communicate……read…..in SCHOOL. And yet English classes are seen as the most irrelevant of all by STEM folks.
Now some of you still got those apostrophes wrong but you are forgiven. I mean that is not the point. The point is that we learn/teach things that perhaps may seem ridiculous in high
school, but we if are going to remain an educated culture that has vibrant communication (like this thread), some apparently irrelevant subjects are indeed utilitarian.
And forget Battle of Hastings; I am certain your history teacher was aiming a bit higher than that.
I struggled with this for a long time as an elementary and middle school math teacher. Eventually, I realized two things:
1) Character traits are often more important to success than particular knowledge/skills. Grit is something that you learn by working through hard problems.
2) Interesting != Useful. The best teachers help students see what’s intrinsically interesting about a subject. Once students are engaged, the question, “When are we going to use this?” doesn’t even arise.
A teacher who has thought deeply about this in math education is Dan Meyers: http://blog.mrmeyer.com/ Although he focuses on high school education, I think his ideas would work at the college level as well.
Character traits are inborn, or at least beyond our control. You’re simply rewarding or punishing students for being who they are. You should kill yourself
Two responses:
1) Yup, we probably overrate how much good learning “useless” subjects will do to students, and probably overestimate the harm that happens by doing it badly (or perhaps at all).
2) Transfer is a slippery cognitive phenomenon. Oftentimes it does indeed look like it almost never happens, except when there’s much overlap in situational features. On the other hand, there’s something to be said about the alternative “preparation for future learning” view of transfer, where transfer isn’t just literally using the knowledge/skill bits you learned from before in a new situation, but your learning in that prior situation having better prepared you to ask the right questions and use the resources in your new situation to achieve understanding/solve your new problem.
See here: http://aaalab.stanford.edu/transfer_and_learning/tr_preparation.html
and here:
Bransford, J. D., & Schwartz, D. L. (1999). Rethinking transfer: A simple proposal with multiple implications. Review of Research in Education, 24, 61-100.
3) From the perspective of creative problem solving, I think it is helpful to cultivate interests and skills in a heterogeneous set of domains. It can give you a wider range of representations and solution approaches when you’re faced with a difficult problem. However, I agree with you that this needs to be intrinsic-interest-driven.
Even if transfer learning would not work at all, we should teach people a wide array of things in school. Many young people don’t want to learn things that they won’t use in their future job. But they don’t know what job they will be able to get in the future (the job market is complicated and unpredictable), or what skills are useful there. Further, they won’t know if they want to work with skill/subject X until they have learned enough about it to understand.
Finally, even if transfer learning does not work at all, knowing many different things will certainly aid a particular type of creativity, where you come up with new facts or new things by combining disparate previously known facts.
In Italy, we have a high school (Liceo Classico) where we study Latin and Ancient Greek. I attended it and I finished my course with top marks (so I know what I am talking about). People say Latin and Greek are useful to form your mind and, as evidence of this, they say that former Liceo Classico students perform very well at university. But… they do not say that these kids are always good students also before attending Liceo Classico (which is actually a difficult school) and often their parents have a higher education and sometimes their families are wealthy. Thanks for your post.
@Julian Togelius
I’d go along if I thought that we picked teaching topics based on what people are likely to need later. I don’t think it works this way. For example, we teach advanced trigonometry, but rarely teach probabilities in high school.
Few will ever need anything from trigonometry but the basics. Heck! I am a college professor with a Ph.D. in engineering mathematics and I don’t recall the last time I had to use a trigonometric identity, let alone some of the crazy ones. And if I needed to do trigonometry today, I would probably not bother with the crazy identities. Is it our belief that people (what proportion… 0.1%?) who will need to know these advanced trigonometric identities won’t be able to learn it then? Are we so arrogant as to think that all should learn it so that the few who ever need will know them?
I don’t think it is the logic we use. We impose trigonometry to all students on the basis that even if they never use it, it will be good for them to do some crazy mathematics.
(And also because they will need to differentiate cos^2(x) later if they take calculus… but why exactly do they need to differentiate cos^2(x)? Hardly a useful skill for most people…)
(BTW the net effect of these teaching strategies might well be negative as it can be shown that most students finish high school with a hatred of mathematics. So if they do need trigonometry later, they may reasonably try to avoid it for fear of boredom. Hardly what we set out to do…)
I’d like to read more of the research on the subject of transfer. For example do the authors discover that it never occurs? Or that ‘on average it has little effect’? If the first, then I believe it is still valuable because (for example) you cannot say in advance that (e.g.) trigonometry will have no use to someone studying encryption techniques. Or that studying psychology will have no value to a programmer.
While in the general case there may well be no cross-pollination across such fields, in some specific cases the crossover may produce startlingly original insight, and I think this is where huge value can lie.
In my own experience of 30-odd years of programming I have found very valuable crossover from my previous career as an electronics technician. And in that earlier career I recall being significantly surprised as I drew on my high-school math experience with imaginary numbers – surely useless knowledge indeed!
I think while you are making the point (and quite well!) for one way the other side is quite important. It is confusing that learning the “other” things would help in critical thinking, it is not necessary. It is not really clear what helps in critical thinking, indeed had that been know then that is what would be taught. My understanding is that when one does develop or starts to develop critical thinking – then all the “useless” things whether done for fun (more like to help) or imposed help and help a lot in a non-tangible non-obvious way.
I’d argue there are useful transfers from trigonometry. It’s a helpful tool for approaching Non-Euclidean geometries, which may end up being the universe’s geometry and at the very least define distances on Earth.
Additionally, the trigonometric functions encapsulate discontinuous and periodic functions rather nicely. If you can understand function composition and calculus involving trig functions, you’ve gone a long way to understanding function composition/calculus in general. And I’ve certainly found ample opportunities to reason about how different processes would combine or change over time.
Vector space models can be great for reasoning or calculating on complex objects as a class, rather than reasoning from the properties of each variable explicitly.
You certainly don’t need to be able to recite the trig identities, but I’ve found the intuition about angles has stuck with me and come in handy. On the same tokens I’m not an economist but a basic handle on the workings of an economy help me understand the connections of what I learn about the world around me.
I think that arguing about why (say) trigonometry might be useful misses the point. We can construct a plausible rationale for all knowledge being useful – after all, that’s why it first started being developed !
The point is that given the limited amount of time that students in school have, what are the MOST important things they should learn from any topic. For example, there’s a really strong argument for probability based on just the sheer number of places where a misunderstanding of probability can cause problems. This is a feature of our modern data-driven world, and wouldn’t have been true 100 years ago. And while geometry is one of my favorite topics, I don’t necessarily see the need to use it as “the way to expose students to logical reasoning”. In fact, a study of discrete mathematics or graph theory might do the same, and is easily relatable to student experiences.
I’d be willing to accept the meat in your argument — not everything you learn is useful per se, and given our limited time, we’d do well to try to be selective, if we can. But I’d say that even useless learning has its purpose in keeping the brain “alive”. It’s a bit like gymnastics/exercising: not every kind is useful for you, and some of them may even cause harm if done wrong. But in general any exercise is better than doing nothing. As for the brain, though not totally proven it appears than making it “exercise” would be good to prevent cognitive degeneration in old ages (though that is mostly unrelated to learning at early ages).
And I’d take issue with you example: mathematics *is* useful/important for everybody. Though you later qualify it in one comment: from among branches of mathematics I agree to stress probability, statistics and basic calculus, and less other areas (such as differential calculus or trigonometry). Numerical literacy for everybody would help to avoid people getting tangled in mortgages they don’t understand, or fooled by bogus statistics (the famous “How to lie with statistics” book should probably be compulsory 🙂
There are learning experiences that can leave a lasting impression without imparting anything useful. Consider for example an engineering student who takes an art history class. Let’s not make a fetish of utilitarianism.
Can anyone here argue in good faith that we should be privileging trig over probabilities in high school?
As to what is useful, the goal is to form critical thinkers that can be effective workers and good citizens. I’m pretty sure we’re failing at all of those tasks right now.
As Sherlock Holmes said:
‘I consider that a man’s brain originally is like a little empty attic, and you have to stock it with such furniture as you choose. A fool takes in all the lumber of every sort that he comes across, so that the knowledge which might be useful to him gets crowded out, or at best is jumbled up with a lot of other things, so that he has a difficulty in laying his hands on it. Now the skillful workman is very careful indeed as to what he takes into his brain-attic. He will have nothing but the tools which may help him in doing his work, but of these he has a large assortment, and all in the most perfect order. It is a mistake to think that little room has elastic walls and can distend to any extent. Depend uon it – there comes a time when for every addition of knowledge you forget something that you knew before. It is of the highest importance, therefore, not to have useless facts elbowing out the useful ones.’
I attended Liceo classico in Italy like Marco. I’ve to confess that for me studying Latin and ancient Greek was fun, and what I learned is still useful for me today.
When learning languages (dead or alive, it doesn’t matter) there is some sort of critical mass: when you have acquired a good knowledge of say four of five different languages, everything suddenly becomes easier. The same holds for math, up to a certain degree. The key point is that you need a deep understanding of the subject; today’s skill oriented learning is not enough to trigger the advantages of being a polymath.
Stating that learning useless stuff is useless is pretty much beggin the question. Of course it is – by definition.
Learning Latin (as I did) is valuable in learning formal grammar rules (which are seldom taught when one learns english), so learning other languages becomes easier, and one has a language to use for understanding other grammars (including computer languages). It is also extremely valuable in learning French, Spanish and Italian – all of whose newspapers I can read.
Another example is Music. One can argue that music is irrelevant for a scientist, however, there are numerous studies linking music to better performance in math.
Learning the “classics” gives me an appreciation of how little is new in terms of humanities – a broad perspective of the deep problems of mankind, and lessens my hubris as a scientist.
In the workplace, a sales education has helped me be a better programmer by enabling a much better communication channel between our programming department and the front end of our company.
Did I know that any of these things would be useful 30 years ago when I studied them? Not at all. Does anyone really know what they will be doing in 10, 20 or 30 years – and what studies might either directly or peripherally be useful? I sincerely doubt it.
@Stefano, it was fun to me, too. 🙂
I was just saying that the importance of learning difficult things is overestimated.
You say that after a degree everything becomes easier.
I do not completely disagree with you, but are you sure that it is not because you become more familiar to a subject and not smarter?
I do not say that this is a negative thing. Anyway, you improve your understanding. For this reason, I am a believer in deep and vary knowledge. Perhaps, this is what Latin and Greek gave me.
This is a good topic ::
It’s interesting how many successful entrepreneurs started their careers without finishing a formal education …
This seems to confuse learning and teaching (or learning and education).
I’m sure that sitting in a course and having a subject matter (say, projective geometry) poured into your head doesn’t help you with unrelated subjects in the future.
But self-directed learning, where the student is actually active rather than a passive receptacle, I think does help, Figuring out projective geometry for yourself not only teaches you the subject, it is practice in how to figure out complex things.
The real subject matter of an education should be learning how to learn.
“But self-directed learning … ”
Now that’s an entirely different matter — and I for one think it’s the most important factor in learning.
I went to college for comp sci in my late twenties to realize a life long a dream of being a professional java developer …
Yet I had kids — no money — and a lot of bills so I had to put it on the back burner —
A few years later I was working for a company that owned 33,000 web properties. I was in a sales position and wanted to leave because I wasn’t making nearly what I expected to make.
When I gave my notice the CEO looked at my background {which is checkered} and my college scores {pretty good} and asked if I wanted to work as an SEO … I didn’t know what that position really was but I had a pretty good idea …
After a few months at the new position I was speaking with one of the lead developers and told him
that my life’s goal was actually to be a developer … So, I asked him how I could move into application development … He just looked at me and didn’t say anything …
I kinda’ thought I was after his job … so I dropped the subject but a couple days later
he dropped a 1200 page book on ColdFusion in front of me and said, “We need developers – if you want to be a developer – be a developer … ”
That was 2006 … I’ve read a lot of 1200 page books since then and still doing what I love …
But other than a fantastic couple of years I basically been doing this because I love it …
Cause’ I’m definitely not getting rich … But learn? I study non-stop …
My 2 Cents.
I have to imagine that it would be very difficult to test college students for transfer, considering the problems even measuring what they learn. I will say a couple things about the utility of various math topics:
1. If you were an electrical engineer, you might have had more use for trig.
2. Trig is, in may respects, easier to understand and teach than probability. Considering how low paid K-12 teaching is, one shouldn’t be surprised that it generally attracts people who avoided math like the plague in college. Asking many K-12 teachers to teach probability is likely a recipe for disaster. In any event, it’s in geometry, not trigonometry, where you learn logic and proofs.
3. There’s transfer of knowledge and there’s transfer of ways of thinking and of work skills. Math is certainly as useful as introductory CS for learning such things (maybe more so, considering how poorly Intro to CS is taught many places). Clearly, there are some things valuable to the practice of developing software that one can learn while learning math that one won’t learn, or learn as well, in CS0 or CS1. Like systematic, precise thinking.
Finally, while many of these subjects might not make one a better computer scientist, it might make one a better human being.
@Mike Stiber
There are definitively people who need to know the identity sin(2u) = 2 cos(u) sin(u). But it is a good bet that it is not much more than 1% of all people. Maybe learning this identity makes you a better human being. Maybe.
I certainly didn’t remember that identity. But there’s so much to sin(2u)=2 cos(u)sin(u)!
Just to start, imagine sin(u). sin(2u) is going to compress that into half the period. That means twice the zero’s. And of course, cos and sin have different zeroes and the same period, so multiplying them would also give you double the zeroes. But isn’t it interesting that the zeroes of cos and the zeroes of sin correspond to alternating zeroes of sin(2u)? And here we can start to see how the fact that the angles in a triangle sum to pi/2 and the definitions of sin and cos in terms of ratios of sides leads to sin(x)=cos(pi/2-x) and that leads to the even spacing of the zeroes. I could go on about where the 2 comes from…
That reasoning uses isomorphisms between a graph, an equation and a definition about angles and triangle sides. What was easy to state in an equation was easy to reason about with a mental graph which added insight to a geometry. I’m convinced that intuition about isomorphisms are hugely valuable for general reasoning, and that being able to switch between them in trig transfers to the isomorphisms in other math fields (e.g. slope, derivative and velocity). Over a decade after a trig course I don’t know many identities but can still reason about what they say. And there are isomorphisms entering and leaving trig: from that graph I could start thinking about integrals to get areas, from the area I could think about distance. And in the other direction the triangle could be the angular distance between two documents I’m thinking about in terms of a hyperspace embedding.
I will agree that the way we usually teach children is to just have them memorize that statement as a fact. But I think there’s very little worth memorizing by rote, and very little not worth finding isomorphisms to other portions of your collected knowledge.
@Daniel: It might be more than 1%, but even if it wasn’t, would you be able to tell which 1% before kids even finish high school, knowing that those who didn’t would be highly unlikely to ever become engineers?
@Mike Stiber
Almost everyone needs to understand the basics of probabilities. Enough to understand a throw of dice. Yet we do not teach anything about probabilities in high school (at least where I live).
You know how I learned probabilities? By playing D&D. I had to figure out whether 2D6 was better than 1D10. I just ran the numbers and came up with an answer.
How many kids can figure out the probability of getting a 7 with 2D6? Probably only the top 5%.
It is pretty pathetic.
I know I sound very utilitarian and even maybe a tad red neck, but I think we should focus on teaching useful stuff.
Was it Alfred the Great vs. William the Conqueror – 1066?
Unfortunately you (and I) mostly had bad history teachers.
There could be an entirely different reason for why there appears to be no transfer and indeed not a small amount of anger about “useless stuff”.
This comes back to my personal theory that about 15% of teachers are very good, 70 percent are mediocre and 15% are downright awful. We each have a decent teacher of some subject or other in our formative years. We also most likely have a terrible teacher of some other subject in those formative years. We decide, based on these two instances, that one subject is great and will lead towards our future studies and careers, and the other one is useless.
I argue that it is the emotional bias that prevents any skill transfer, mostly because in a class we actively dislike or feel we suck at, we DON’T LEARN ANYTHING – so there is nothing to transfer. If we love a class and are good at it (usually the same thing), that knowledge transfers into EVERYTHING.
I agree that there is almost zero transfer between skill set training. It’s simply what the research shows. (I even think the French/Spanish connection is probably weak or non-existent.)
Furthermore, given that there’s no transfer and that most of what is learned is forgotten, the vast majority of classroom time is wasted.
For example, pretty much every moment spent in history class is a total waste. There is no “skill” transfer, and nobody remembers any of the information anyway. As proof, perform this simple test. Go up to 10 college-educated people and ask them who were the combatants in the Battle of Hastings. My guess is *maybe* one of them will know. Probably 0.
@Matthias
It is true, but as Steve Jobs noted:
“A lot of people in our industry haven’t had very diverse experiences. They don’t have enough dots to connect, and they end up with very linear solutions, without a broad perspective on the problem.”
It is very hard to tell what knowledge is useful and what is lumber.
Hello,
I would just like to point out that every single one of the respondents learned how to write…..communicate……read…..in SCHOOL. And yet English classes are seen as the most irrelevant of all by STEM folks.
Now some of you still got those apostrophes wrong but you are forgiven. I mean that is not the point. The point is that we learn/teach things that perhaps may seem ridiculous in high
school, but we if are going to remain an educated culture that has vibrant communication (like this thread), some apparently irrelevant subjects are indeed utilitarian.
And forget Battle of Hastings; I am certain your history teacher was aiming a bit higher than that.
Ann
English Teacher